I'm not a logician at all, but since I'm using categories, I decided at some point to find out what is going on. A logician would probably give you a better answer, but in the mean time, here is my understanding.

Grothendieck's Universe axiom (every set is an element of a Grothendieck universe) is equivalent to saying that for every cardinal there is a larger strictly inaccessible cardinal. 

(Recall that a cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of cardinality $<\lambda$) and b. for all cardinals $\mu<\lambda$ we have $\mu^+<\lambda$ where $\mu^+$ is the successor of $\mu$. Strongly inaccessible cardinals are defined in the same way, with $\mu^+$ replaced by $2^\mu$. Usually one also adds the condition that $\lambda$ should be uncountable.)

Assuming ZFC is consistent we can show that ZFC + " there are no weakly inaccessible cardinals" is consistent. The way I understand it, this is because if we have a model for ZFC, then all sets smaller than the smallest inaccessible cardinal would still give us a model for ZFC where no inaccessible cardinals exist. See e.g. Kanamori, "The Higher Infinite", p. 18.

So far the situation is pretty similar to e.g. the Continuum Hypothesis (CH): assuming ZFC is consistent, we can show that ZFC+not CH is consistent.

What makes the Universe Axiom different from CH is that we can not deduce the considtency of ZFC + IC from the consistency of ZFC (here IC stands for "there is an inaccessible cadrinal"). This is because we can deduce from ZFC + IC that ZFC is consistent: basically, again all sets smaller than the smallest inaccessible cardinal give a model for ZFC. So if we can deduce from the consistency of ZFC that ZFC + IC is consistent, then we can prove the consistency of ZFC + IC in ZFC + IC, which violates G\"odel's incompleteness theorem (see e.g. Kanamori, ibid, p. 19).

Notice that the impossibility to deduce the consistency of ZFC + IC from the consistency of ZFC is stronger that simply the impossibility to prove IC in ZFC.

So my guess would be that Cartier is referring to the fact that the consistency of ZFC implies the consistency of ZFC plus the negation of the Universe Axiom, but not ZFC plus the Universe Axiom itself.